Glass mirror



F. A. BENFORD GLASS MIRROR Filed July 2, 1922 3 Sheets-Sheet 1 Inventor Frank A. E g/bra,

His Attorney F. A. BENF'ORD GLAS S MIRROR Filed July 22, 1922 3 Sheets-Sheet 2 I ANGLE a 9 4,

ANGLE g Inventbr: E ankA Belfrd by M/M His Attorney July 8 1924. 1,501.03!

F- A. BENFORD GLAS S MIRROR Filed July 22, 1922 3' Sheets-Sheet 3 Inventor Eank A Behfbfd,\

His Attor may Patented .luiy 1 24.

NT FFIE.

FRANK A. BENFORD, OF SCHENECTADY, NEW YORK. ASSIGN'OR Tb GENERAL ELECTRIC COMPANY, A CORPORATION OF NEW YORK.

GLASS MIRROR.

Application filed July 22, 1922. Serial No. 576,686.

5 State of New York, have invented certain new and useful Improvements in Glass Mirrors, of which the following is a specification.

My invention has reference to mirrors in general. More in particular, it has reference to mirrors that are used as reflectors for searchlights or in connection with zip-- invention has special reference to mirrors of the paraboloida-l type. In practice, it is found that a glass paraboloidal mirror. in a Searchlight for example, gives two beams, one from the front surface of the mirror and one from the back or silvered surface. The former beam carries, roughly speaking. only five per cent of the total intensity of the light, while the beam from the back surface carries the remainder of the light, or about ninety-five per cent. It is found in practice that these two beams do not coincide. It is one of the objects of my invention to make a mirror that will cause these two beams to coincide in order to produce a more efficient light and a beam of greater sharpness. If a mirror is used in which the front surface is a paraboloidal surface, I find that the second or silvered surface must be so formed as to compensate for the angles of refraction of the light which is reflected from the silvered surface. This refraction takes place at two points, namely-at the point where the light enters the glass, and the point where the light leaves the lass after having been re flected from the iiack or silvered surface. I find that such a back surface is not parallel to the front surface, nor is it paraboloidal in form. It does, however, approximate the paraboloidal form.

Another object of my invention is to pro vide other details of improvement tending to increase the efiieiency and serviceability of mirrors of the above character.

It is a matter of considerable difficulty and a matter of considerable labor to determine the rear surface of mirrors of the above character and I shall proceed now to describe how such surfaces are obtained in the accompanying specifications, reference being made to the accompanying drawings.

Referring specifically to the drawings, Fig. 1 represents an axial section of the mirror constructed in accordance with my invention. In this figure, I also show onlarged the diagram of the course of the rays of light which originate at the focal point and which are reflected by the mirror.

Fig. 2 is a curve showing the relation between the angle 12-0. and the angle a of Fig. 1.

Fig. 3 illustrates the thickness of the mirror at the limiting angle of 180 degrees.

Fig. 4 is a curve showing the relation between the thickness of the mirror along the w-axis and the angle (1+1) 2 of Fig. 1.

In Fig. 5, a diagrammatic comparison is made between two different methods of solution on a scale of 20 to 1. Also an enlargement of a section of the diagram on a 1000 to 1 scale.

Fig. 6 illustrates the triangle method of solution.

I will now refer more in detail to the drawings in connection with which I will make use of a series of symbols. The following is a tabulation and descriptionof these symbols:

'Fzfocal length of the paraboloidal front surface. The common method of making. F:1 will be followed so that F becomes the unit of measure and disappears from the equations.

T zthe thickness of the glass on the axis of projection,i. e., along the m-axis.

T ztlle thickness of the glass at some other angle a from the axis of projection.

nzthe refractive index of the glass as compared with that of air taken as unity. In this computation 2021.52.

(X Y,):the rectangular coordinates of a pointon the paraboloidal front surface, referred to the vertex of this surface as origin of coordinates. This point (X Y is taken as the point of entrance of the ray leaving the focal point at the angle 0:.

(X Y,):the rectangular coordinates of the point at which the ray through (X 1 emerges from the glass.

(X Y ):the rectangular coordinates of the point at which the ray through (X 1 is reflected by the silvered back'surface of the mirror.

(X Y,):a point in a plane placed perpendicularly to the axis of projection at a distant X from the origin.

(X Y ):the point where the bisector of the ray path in the glass intercepts the front surface of the mirror.

azangle between the incident ray through the point (X Y,) and the :c-axis.

g=angle between the incident ray and the normal to surface at point (X,Y,).

bzangle between the radius vector from focal point to the point of emergence (X 3 and m-axis.

=angle between the emergent ray and the normal to front surface at point c angle between normal to front surface at point (X Y,) and the path of the refracted ray through the glass.

d=angle between the path of the refracted ray at the point (X,Y and the a axis.

e angle between the normal to the front surface at the point (X Y and the path of the emerging ray through the glass.

f=angle between the paths of the ray through the point (X Y before and after refraction.

p length of radius from focal point.

The familiar optical rules relating object, radius of curvature, and image do not readily apply to the problem in hand. The curvature of the mirror decreases rapidly as we move out from the axis so that any investigation starting with radius of curva' ture must necessarily soon become quite involved. There is one method of attack that is rather simple and while a considerable amount of labor is involved in carrying it out, the results are certain and exact. This is the method of equal optical paths or equal times, and is here used in determining points on the second or back surface.

Two methods of computing the desired back surface will be given. One is the above mentioned of equal paths and the second We will call the triangle method, to be later described. The first method will be used as a standard to which solutions 'by another method herein described, will be referred.

Let us consider a point source of light placed at the focus of the first surface, Fig. 1, to give off a single pulse of light. This pulse will advance in all directions in a vector, measured spherical wave until it reaches the mirror. One section of the wave, which we may call a ray, moves along the w-axis, meets the front surface of the glass at O and passes into the glass, reaching the back or silvered surface at the point 0 At this point the ray is reflected back along the same axis and reaches the plane X at the point (avg As pointed out, the path from O to O an d back to the plane X along the w-axis is partly in air and partly in lass. We know that the rate of travel of lig it through glass is n times more slow'than in air. Therefore, the distance L that the ra would travel in the same time through air is obviously We know that if the light is reflected by the front surface the reflected rays will all be parallel to the ac-axis inasmuch as the curve of the front surface is paraboloidal. The condition set for our reflector is that the two beams shall coincide and thereforeemerging ray at the point (00 3 must also run parallel to the .T-axis. This condition will be fulfilled when all rays that leave the focal point and that are reflected by the back surface, take the same time to reach the plane X This means that the distance L that any one of the rays travels from the focal point to the back surface and to the plane X, shall be The next step is to find the itemized statement of L similar to that of L in equation 1, and equate the two itemized statements.

The general method of procedure for this step is outlined below 1. Choose any ray making some angle a with the m-axis and find its point of intersection (X,Y,) with the front surface.

2. Trace the path of this my into the glass from the point of entrance (00 3 to the point of reflection (00 3 and find the equation of the straight line L,, representing this path, which is as yet of an unknown length.

3. Select some point (m g on the front surface at which it is graphically or otherwise estimated, as a first approximation, that the ray will emerge and move parallel to the w-axis.

4. Trace the path of this ray into the glass from the point (03 and backward through the glass to the point of reflection (ag y and find the equation of this straight line path designated as L and which is as yet of an unknown length.

5. Find the intersection of the two paths L, and L as a trial point for the point of reflection (a y on the back surface.

6. Find the total length (LA-L of that portion of the path of t e ray confined within the glass, and substitute in the itemized statement of L, noting whether the path thus obtained by trial is too short or too long to satisfy the equation of equal paths.

7. Repeat operations 3, 4, 5 and 6 until the equation of equal paths is satisfied with the desired exactness.

The foregoing having been. set out preliminarily, we will now proceed with the operation of ascertaining the items for the length of path L (equation 2) in order to equate with the itemized statement in equation 1.

Consider the ray that enters the glass at point (avg This ray is reflected at the point (m g and emerges again at the point (40 3 striking the plane X at the point (21 3 From the focus to the point (00 3 the optical path is F--I-. z2 which follows from the usual definition of a parabola as being the locus of a point equally distant-from a fixed point and a fixed line. Denoting by L the actual distance between points (00 3 and (00 3 and by L the distance between (00 and (00 3 the equivalent length in air of this optical path through the glass is n(L,+L After the second refraction at the point (w g the path, to the plane X, is (m -m Adding the separate items of the path we get but (1) and (2) are to be equal, hence o+ 4 1+ 1+ 2) b h 2 Whenequation 3 is satisfied for all points on the back surface, then the reflected as originthe focal length F being taken as unity. The polar form of expression for the same curve is 1; cos 2 Where P is the radius vector, or distance from the focus to a point on the curve.

Selecting some convenient angle a, we have yp=2 tan 2 (6) and The angular relation between the incident and refracted ray at (avg is and the equation of the line passing through (gag 1) and (m g may be written immedlate y y,=m tan d-l-C (10) leaving only G to be determined by substituting numerical values of my, and tan d in the equation.

This is as far as the work can be carried by a direct process for the position and direction of the reflected ray L cannot be directly determined without knowin in ad-- vance the point of reflection (m y The next step is to estimate the location of the point (02 1 on the front surface at which the ray through (mg is going to emerge. In computation of this sort the accuracy of the result is usually required to be as great as possible using a seven place table of logarithms, which inthis particular work will give an error of one part in about one hundred thousand. It is thus evident that an estimate of the emerging point (93 will give highly inaccurate data, but such data may be used as a basis for a second,

third, etc., estimate that will gradually ap proach exactness.

Having selected some point- (02 as worthy of trial, the ray is traced backward through the glass'to find the crossing point (a p with the path through (w g To determine the angle between the normal at (w g and the emergent ray, we have n sin e=sin g- (12) and the angle f between the path L and the ray leaving (w y parallel to the axis is also &

and the equation of the line of the path L is 2 /2 2 tan L M cos d (16) and the length of L is z= (josf 7e are now in possession of all data necessary for a trial solution of equation (3). If the right hand member of (3) exceeds 211T in value, we select a smaller value of 3/2 for the next trial, or, if the path is too short, we take a greater value and repeat the opgration indicated by equations (11) to I21 checking the design of a mirror that is finished one of the simplest ways is to measure the thickness from front to back surface at various points. This is easily done with great exactness on a broken mirror for then small micrometers reading to 0.0001 inclusive may be used.

.It should be pointed out that in using micrometers the line of measurement is not normal to either of the non-parallel surfaces, but it is the least distance between a given point on one surface and the other surface. To get this line of least thickness by a method of computation would be a lengthy process and the extra work would not be justified by the small gain in accuracy.

y x tan C (18) s a where m= ten (19) The intersection of this normal with the parabolic front curve is obtained by solving al ilih 20 w 3 of the lines of.

the thickness of glass is an -x Equation (22) is the more accurate for points within 60' degrees of the axis; beyond this point (23) is the better.

It has been found that operation (11) to (17) must be repeated from four to six times to arrive at a fairly exact value for (00 3 Here lies the objection to this method. Each point (w g determined on the back surface requires some 400 opera, tions with seven place logarithms for the solution of equations (6) to (10) once each and (11) to (17 about five times each. If it is the purpose of the designer to make a templet of the back surface it may be necessary to solve for a point every half inch along its length. For a 60 inch diameter mirror, this would mean 60 solutions, or about 24,000 operations with seven place logarithms. The cost of such a series of computation is in many cases prohibitive.

To overcome this difficulty a direct methodof solution has been developed. This method, while derived from certain optical laws, is empirical in nature and the question of its accuracy immediately arises. This method, which is outlined below, has been checked against the exact method in a number of ways and found to give data of sufficient accuracy for the manufacturer to work to.

Let us assume that a mirror is built with a great depth along the axis of projection, so that the two branches of the generating parabola become nearly parallel to the axis. The light from the focal point will strike the surface at the distant edge in a direction close to the grazing angle, and after refraction, reflection and a second refraction the light will travel parallel to the axis without much change of direction. Fig. 3 represents roughly the condition at the edge of a very deep reflector when the generating angle a approaches the limit 180 de es.

The altitude of the trian le i 3 formed y the points (ma, 5%) is nearly equal to the thickness of the glass because the curvature between (w y and (w y is small and the difference between the curve and a. straight line is not great.

Measuring from (w g to (w y the optical length of path is and also the triangle being isosceles,

The base of the triangle has a length B=2T tan 0 (26) and from the construction of the figure it is evident that ac -x 221 tan 0 sin 1a (27) Substituting these values in equation (3) cos c (nsin 0 sin %a) (28) but sin a=n sin c (29) and equation (28) becomes cos c 2nT cos c (n 1:. sin 0) cos c (30) 2nT cos c or, rearranging,

T.= To

COS 0 E nation (31) although developed solely as a 'ting case, is fortunately a very good approximation at all points between a:0

and a:180, as is shown in the tabulation below. The calculations are based on the angle a at which the ray enters the glass, and equation (31) may be rewritten in a slightly more convenient form by changing from angle 0 to angle 1a T,,= To

sin {ca 2 X,-=F tan *Gr" o Thickness of mirror.

29 2%? Thickness 0 mm from (31) 5. 2 tats M33) Parametric equations for points on back surface.

The triangle of equation (31) may be used to locate (w y with a degree of accuracy that is surprisingly good. Referring to Fig. 6, we can write T cos c (34) for, according to construction the bisection T is parallel to the normal at a on the surface. Substituting T for T, from (31) o cos c (35) Again referring to Fig. 6, it is evident that m '-:c =S cos (a+c) 1 a'- /1= where (w' y' is the (w y point as determined by the triangle method.

In-equation (36) we may substitute from the following relations in order to eliminate w g and c and have only the parameter a (or 1a) and the constants F and T sin (1}a+c):sin l a cos cl-sin 0 cos 4a (39) cos (1};a+c) :c'os 1a cos csin 1a sin c (40) m :F tan 4a (41) ::2F tan 1a (42) sin c= tion (36) may be written The solution of (43) is considerably simpler than the solution of the path equation (3), not only because a single solution is' 0-0. 0382369-0. 0382369 0 100. 030307 7-(]. 0303082 0. 1805068 20 0. MIG-0. 0060443 0. 3636953 30+0. 0360576+0. 0360518 0. 5524151 0. 5524181 40+0. 0987078-H). 0986974 0. 7498718 0. 7498786 50+0. 1852497-Hl. 1862354 0. 9598681 0. 9598809 60+0. 3053371-1-0. 3053198 1. 1871401 L 1871602 Ya Y 3 In Fig. 5 a section of the mirror at 50 is shown enlarged twenty times, with two triangles showing the solution by equal paths (full lines) and the triangle method (dotted lines). From the figure it is seen that the triangle gives 1. Direction of entering ray correctly.

2. Location of apex of triangle fairly exactly.

3. Height of triangle nearly equal to length of bisector.

4. Reflected ray from w y in error by about 1.

5. Refracted ray in error by 1 30.

The method is thus not suited for tracing a ray completely through the glass, or for finding the slope of the back surface, but these two features are not of great importance in the practical design of a mirror.

To complete the solution, it is necessary to find the point on the front surface to which the solutions of (33) and (43) apply. This point may be determined by getting the intersection of the front parabola by a line arallel to the normal at angle a. It will e noted that this does not quite agree with the bisector used in the original solution. This disagreement is not a serious matter, as it has small effect on the calculated value of T...

The line through (m' 3/ has for its equation y s 3 tan 5 where only C remains to be evaluated.

Equations (18) and (20) are'then used, but instead of (19) we have M= tan la (45) Although I have illustrated my invention in connection with a reflector, the front surface of which is paraboloidal in form, I wish it understood that my invention is not confined to a paraboloidal type of reflector but a plies as well to other forms such as an e hpsoidal and hyperboloidal and the like.

In the foregoing, I have first described the direct method, that is, the equal path method and have used it as a check upon two approximation methods, namely, the triangle method and the limiting values method. The direct method may also be called the exact method. The approximation methods therefore may be. used in making the reflector of my invention since the maximum error when either approximationmethod is used falls within a small fraction of 1 per cent of the result when the exact method is used as indicated by the table in page 13. What I claim as new and desire to secure by Letters Patent of the United States, 15:

l. A glass reflector having a rear surface and having a paraboloidal front surface, the former so located with reference to the front surface and of such configuration that the thickness of the glass as measured along any line normal to the rear surface is substantially where a is any angle.

2. A glass reflector having a rear surface and a curved front surface, the latter having a major axis and a focal point therein, the back surface so located with reference to the front surface and of such configuration that the thickness of the glass as measured along a line normal to the rear surface is substantially where a is any angle and where 0 is the angle made by the line of refraction of a' ray of light from the focal point of the front surface and where the ray enters the glass same ray as a reflected ray rom the rear surface is *cos d-Z where a is any angle 4. A glass reflector having a rear surface and having a paraboloidal front surface, the former so coated with reference to the front surface and of such configuration that the thickness of the glass as measured along a i it line normal to the front surface is substanfamily cos c where 0 is the angle between the direction of a refracted ray incident from the focal point of the front surface and a normal line to the front surface at the point where said incident ray strikes the front surface and where T is the thickness of the glass along the major axis and where a is the angle of the said incident ray with the major axis.

6. A glass reflector having a rear surface and a curved front surface, the latter having a major'axis and a focal point therein, the back surface so located with reference to the front surface and of such configuration that the thickness of the glass as measured along any line normal to the front surface and piercing the back surface at the point of reflection is substantially cos c where c is the angle between the direction of a refracted ray incident from the focal point of the front surface, and a normal line to the front surface at a point where said incident ray strikes the front surface and where T is the thickness of the glass along the major axis and where at is the an 1e of the incident ra with the ma or axis.

A less reflector aving a rear reflecting surflace and a front paraboloidal surface, theformer so located with reference to the front surface that a ray radiated from the focus and at any angle a from the major cos c 8. A concavo-convex glass reflector hav- 111% a paraboloidal front surface and a rear ,re ectmg surface of revolution, the thickness of the glass between the two surfaces measured along any line perpendicular to the front surface being within 1 per cent of the thickness necessary to reflect every ray incident from the-focal point of the front surface in a direction parallel to the major axis of the front surface.

9. A concavo-convex glass reflector havin a araboloidal front surface and a rear re ectmg surface of revolution, the thickness of the glass between the two surfaces measured along any line perpendicular to the front surface being within .01 per cent of the thickness necessary to reflect every ray incident from the focal point of the front surface in a direction parallel to the major axis of the front surface.

10. A concavo-convex glass reflector having a paraboloidal front surface and a rear reflecting surface of revolution, the thickness of the lass between the two surfaces measured a ong any line perpendicular to the front surface being within .002 or cent of the thickness necessary to refl ct every ray incident from the focal point of the front surface in a direction parallel to the major axis of the front surface.

11. A glass reflector having 'a rear surface and a front surface, the front surface being a conic and the rear surface being non-conic, the thickness of the glass between the two surfaces measured along any line normal to the front surface being substantially FRANK a. BENFQRDQ 

